Rust f64::atanh

Percentage Accurate: 100.0% → 99.7%

Time: 4.6s

Alternatives: 6

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \tanh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (atanh x))

C

double code(double x) {
	return atanh(x);
}

Python

def code(x):
	return math.atanh(x)

Julia

function code(x)
	return atanh(x)
end

MATLAB

function tmp = code(x)
	tmp = atanh(x);
end

Wolfram

code[x_] := N[ArcTanh[x], $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))

C

double code(double x) {
	return 0.5 * log1p(((2.0 * x) / (1.0 - x)));
}

Java

public static double code(double x) {
	return 0.5 * Math.log1p(((2.0 * x) / (1.0 - x)));
}

Python

def code(x):
	return 0.5 * math.log1p(((2.0 * x) / (1.0 - x)))

Julia

function code(x)
	return Float64(0.5 * log1p(Float64(Float64(2.0 * x) / Float64(1.0 - x))))
end

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(N[(2.0 * x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (*
   (fma
    (fma (fma 0.2857142857142857 (* x x) 0.4) (* x x) 0.6666666666666666)
    (* x x)
    2.0)
   x)
  0.5))

C

double code(double x) {
	return (fma(fma(fma(0.2857142857142857, (x * x), 0.4), (x * x), 0.6666666666666666), (x * x), 2.0) * x) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(fma(fma(fma(0.2857142857142857, Float64(x * x), 0.4), Float64(x * x), 0.6666666666666666), Float64(x * x), 2.0) * x) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.2857142857142857 * N[(x * x), $MachinePrecision] + 0.4), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right) \cdot x\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right) \cdot x\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right) + 2\right)} \cdot x\right) \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 2\right) \cdot x\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), {x}^{2}, 2\right)} \cdot x\right) \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right) + \frac{2}{3}}, {x}^{2}, 2\right) \cdot x\right) \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{2}{3}, {x}^{2}, 2\right) \cdot x\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, {x}^{2}, \frac{2}{3}\right)}, {x}^{2}, 2\right) \cdot x\right) \]

   9. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{2}{7} \cdot {x}^{2} + \frac{2}{5}}, {x}^{2}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{7}, {x}^{2}, \frac{2}{5}\right)}, {x}^{2}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, \color{blue}{x \cdot x}, \frac{2}{5}\right), {x}^{2}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, \color{blue}{x \cdot x}, \frac{2}{5}\right), {x}^{2}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, x \cdot x, \frac{2}{5}\right), \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, x \cdot x, \frac{2}{5}\right), \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   15. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, x \cdot x, \frac{2}{5}\right), x \cdot x, \frac{2}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

   16. lower-*.f64 100.0

       \[\leadsto 0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]

6. Final simplification 100.0%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \]
7. Add Preprocessing

Alternative 2: 99.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x, x, x \cdot 2\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma (* (* (fma 0.4 (* x x) 0.6666666666666666) x) x) x (* x 2.0)) 0.5))

C

double code(double x) {
	return fma(((fma(0.4, (x * x), 0.6666666666666666) * x) * x), x, (x * 2.0)) * 0.5;
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(fma(0.4, Float64(x * x), 0.6666666666666666) * x) * x), x, Float64(x * 2.0)) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.4 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right) + 2\right)} \cdot x\right) \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right) \cdot {x}^{2}} + 2\right) \cdot x\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}, {x}^{2}, 2\right)} \cdot x\right) \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{2}{5} \cdot {x}^{2} + \frac{2}{3}}, {x}^{2}, 2\right) \cdot x\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{5}, {x}^{2}, \frac{2}{3}\right)}, {x}^{2}, 2\right) \cdot x\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   10. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, x \cdot x, \frac{2}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

   11. lower-*.f64 99.9

       \[\leadsto 0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 99.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 100.0%

   \[\leadsto 0.5 \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x, \color{blue}{x}, x \cdot 2\right) \]

8. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x, x, x \cdot 2\right) \cdot 0.5 \]
9. Add Preprocessing

Alternative 3: 99.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* (fma (fma 0.4 (* x x) 0.6666666666666666) (* x x) 2.0) x) 0.5))

C

double code(double x) {
	return (fma(fma(0.4, (x * x), 0.6666666666666666), (x * x), 2.0) * x) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(fma(fma(0.4, Float64(x * x), 0.6666666666666666), Float64(x * x), 2.0) * x) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.4 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right) + 2\right)} \cdot x\right) \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right) \cdot {x}^{2}} + 2\right) \cdot x\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}, {x}^{2}, 2\right)} \cdot x\right) \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{2}{5} \cdot {x}^{2} + \frac{2}{3}}, {x}^{2}, 2\right) \cdot x\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{5}, {x}^{2}, \frac{2}{3}\right)}, {x}^{2}, 2\right) \cdot x\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   10. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, x \cdot x, \frac{2}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

   11. lower-*.f64 99.9

       \[\leadsto 0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 99.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]

6. Final simplification 99.9%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \]
7. Add Preprocessing

Alternative 4: 99.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 0.3333333333333333, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (* (* x x) x) 0.3333333333333333 x))

C

double code(double x) {
	return fma(((x * x) * x), 0.3333333333333333, x);
}

Julia

function code(x)
	return fma(Float64(Float64(x * x) * x), 0.3333333333333333, x)
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{3} \cdot {x}^{2} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} \cdot {x}^{2}\right) + x \cdot 1} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{3}\right)} + x \cdot 1 \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{1}{3}} + x \cdot 1 \]

   5. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \frac{1}{3} + x \cdot 1 \]

   6. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \frac{1}{3} + x \cdot 1 \]

   7. *-rgt-identity N/A

      \[\leadsto {x}^{3} \cdot \frac{1}{3} + \color{blue}{x} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{3}, x\right)} \]

   9. lower-pow.f64 99.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, 0.3333333333333333, x\right) \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.3333333333333333, x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 0.3333333333333333, x\right) \]
8. Add Preprocessing

Alternative 5: 99.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333 \cdot x, x, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fma (* 0.3333333333333333 x) x 1.0) x))

C

double code(double x) {
	return fma((0.3333333333333333 * x), x, 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(Float64(0.3333333333333333 * x), x, 1.0) * x)
end

Wolfram

code[x_] := N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{3} \cdot {x}^{2} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} \cdot {x}^{2}\right) + x \cdot 1} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{3}\right)} + x \cdot 1 \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{1}{3}} + x \cdot 1 \]

   5. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \frac{1}{3} + x \cdot 1 \]

   6. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \frac{1}{3} + x \cdot 1 \]

   7. *-rgt-identity N/A

      \[\leadsto {x}^{3} \cdot \frac{1}{3} + \color{blue}{x} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{3}, x\right)} \]

   9. lower-pow.f64 99.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, 0.3333333333333333, x\right) \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.3333333333333333, x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 0.3333333333333333, x\right) \]
8. Step-by-step derivation

9. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x, 1\right) \cdot \color{blue}{x} \]
10. Add Preprocessing

Alternative 6: 98.8% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 1.0 x))

C

double code(double x) {
	return 1.0 * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 * x
end function

Java

public static double code(double x) {
	return 1.0 * x;
}

Python

def code(x):
	return 1.0 * x

Julia

function code(x)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x)
	tmp = 1.0 * x;
end

Wolfram

code[x_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{3} \cdot {x}^{2} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} \cdot {x}^{2}\right) + x \cdot 1} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{3}\right)} + x \cdot 1 \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{1}{3}} + x \cdot 1 \]

   5. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \frac{1}{3} + x \cdot 1 \]

   6. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \frac{1}{3} + x \cdot 1 \]

   7. *-rgt-identity N/A

      \[\leadsto {x}^{3} \cdot \frac{1}{3} + \color{blue}{x} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{3}, x\right)} \]

   9. lower-pow.f64 99.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, 0.3333333333333333, x\right) \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.3333333333333333, x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 0.3333333333333333, x\right) \]
8. Step-by-step derivation

9. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x, 1\right) \cdot \color{blue}{x} \]

10. Taylor expanded in x around 0

    \[\leadsto 1 \cdot x \]
11. Step-by-step derivation

12. Applied rewrites 99.2%

    \[\leadsto 1 \cdot x \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))